Theorem mobii 2119

Description: Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Hypothesis

Ref	Expression
mobii.1	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
mobii	⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Proof of Theorem mobii

Step	Hyp	Ref	Expression
1		mobii.1	. . . 4 ⊢ (𝜓 ↔ 𝜒)
2	1	a1i 9	. . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒))
3	2	mobidv 2118	. 2 ⊢ (⊤ → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
4	3	mptru 1407	1 ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Syntax hints:   ↔ wb 105  ⊤wtru 1399  ∃*wmo 2083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-eu 2085  df-mo 2086

This theorem is referenced by:  moaneu  2159  moanmo  2160  2moswapdc  2173  2exeu  2175  rmobiia  2737  rmov  2836  euxfr2dc  3005  rmoan  3020  2rmorex  3026  mosn  3730  dffun9  5386  funopab  5392  funco  5397  funcnv2  5421  funcnv  5422  fun2cnv  5425  fncnv  5427  imadif  5441  fnres  5480  ovi3  6199  oprabex3  6335  axaddf  8199  axmulf  8200  frecuzrdgtcl  10798  frecuzrdgfunlem  10805  fsum3  12098